direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40⋊C2, C8⋊8D10, C4.6D20, C40⋊9C22, C10⋊1SD16, C20.29D4, C20.28C23, D20.6C22, C22.12D20, Dic10⋊3C22, (C2×C8)⋊5D5, (C2×C40)⋊7C2, C5⋊1(C2×SD16), C10.9(C2×D4), (C2×D20).5C2, C2.11(C2×D20), (C2×C10).16D4, (C2×C4).79D10, (C2×Dic10)⋊5C2, C4.26(C22×D5), (C2×C20).88C22, SmallGroup(160,123)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C40⋊C2
G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b19 >
Subgroups: 280 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×SD16, C40 [×2], Dic10 [×2], Dic10, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, D20 [×2], C22×D5, C40⋊C2 [×2], C2×D20, C2×C40⋊C2
►On 80 points
(1 77)(2 78)(3 79)(4 80)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 77)(2 56)(3 75)(4 54)(5 73)(6 52)(7 71)(8 50)(9 69)(10 48)(11 67)(12 46)(13 65)(14 44)(15 63)(16 42)(17 61)(18 80)(19 59)(20 78)(21 57)(22 76)(23 55)(24 74)(25 53)(26 72)(27 51)(28 70)(29 49)(30 68)(31 47)(32 66)(33 45)(34 64)(35 43)(36 62)(37 41)(38 60)(39 79)(40 58)
G:=sub<Sym(80)| (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58) );
G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,77),(2,56),(3,75),(4,54),(5,73),(6,52),(7,71),(8,50),(9,69),(10,48),(11,67),(12,46),(13,65),(14,44),(15,63),(16,42),(17,61),(18,80),(19,59),(20,78),(21,57),(22,76),(23,55),(24,74),(25,53),(26,72),(27,51),(28,70),(29,49),(30,68),(31,47),(32,66),(33,45),(34,64),(35,43),(36,62),(37,41),(38,60),(39,79),(40,58)])
C2×C40⋊C2 is a maximal subgroup of
C8⋊5D20 C8.8D20 C42.16D10 C8⋊D20 C8.D20 D20.31D4 D20.32D4 D20⋊14D4 Dic10⋊14D4 Dic10⋊2D4 D20.8D4 D4⋊3D20 D20.D4 Dic10.11D4 Q8⋊2D20 Q8.D20 Dic5⋊SD16 C20⋊SD16 D20.19D4 C42.36D10 Dic10⋊8D4 Dic5⋊8SD16 C8⋊8D20 C8⋊3D20 C40⋊21(C2×C4) C8.24D20 C40⋊30D4 C40⋊2D4 D4.3D20 C40⋊11D4 C40.43D4 C40⋊15D4 C40.37D4 D4.11D20 C2×D5×SD16 D8⋊11D10
C2×C40⋊C2 is a maximal quotient of
C40⋊9Q8 C20.14Q16 C8⋊5D20 C4.5D40 C23.34D20 D20.31D4 C23.38D20 Dic10⋊14D4 C20⋊SD16 D20⋊3Q8 Dic10⋊8D4 Dic10⋊4Q8 C40⋊30D4
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20H | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 |
| kernel | C2×C40⋊C2 | C40⋊C2 | C2×C40 | C2×Dic10 | C2×D20 | C20 | C2×C10 | C2×C8 | C10 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 |
Matrix representation of C2×C40⋊C2 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 27 | 28 |
| 0 | 13 | 18 |
| 1 | 0 | 0 |
| 0 | 1 | 34 |
| 0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,27,13,0,28,18],[1,0,0,0,1,0,0,34,40] >;
C2×C40⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{40}\rtimes C_2 % in TeX
G:=Group("C2xC40:C2"); // GroupNames label
G:=SmallGroup(160,123);
// by ID
G=gap.SmallGroup(160,123);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations